Controllability is a key concept in control theory. Yet, it does not mean reprogrammability of biological cells. This post discusses the gap between controllability of engineering systems and cellular reprogramming in biology.
References
Y.-Y. Liu, J.-J. Slotine, & A.-L. Barabási, ‘‘Controllability of complex networks,’’ Nature, 473:167–173, 2011.
F.-J. Müller & A. Schuppert, ‘‘Few inputs can reprogram biological networks,’’ Nature, 478:E4, 2011.
B. Fiedler, A. Mochizuki, G. Kurosawa, & D. Saito, ‘‘Dynamics and control at feedback vertex sets. I: Informative and determining nodes in regulatory networks,’’ J. Dyn. Differ. Equ., 25:563–604, 2013.
A. Mochizuki, B. Fiedler, G. Kurosawa, & D. Saito, ‘‘Dynamics and control at feedback vertex sets. II: A faithful monitor to determine the diversity of molecular activities in regulatory networks,’’ J. Theor. Biol., 335:130–146, 2013.
望月 敦史 著，理論生物学概論，共立出版，2021年．
Additional references
S. Gu, F. Pasqualetti, M. Cieslak, Q. K. Telesford, A. B. Yu, A. E. Kahn, J. D. Medaglia, J. M. Vettel, M. B. Miller, S. T. Grafton, & D. S. Bassett, ‘‘Controllability of structural brain networks,’’ Nat. Commun., 6:8414, 2015.
J. G. T. Zañudo, G. Yang, & R. Albert, ‘‘Structure-based control of complex networks with nonlinear dynamics,’’ Proc. Natl. Acad. Sci. USA, 114(28):7234–7239, 2017.
Consider the linear system described by
\[ \begin{align} \dot{x}(t) = Ax(t) + Bu(t). \label{eq:system} \end{align} \]
The definition of controllability introduced by Kalman is as follows.
Definition: The linear system \(\eqref{eq:system}\) is controllable if for every initial state \(x_0\) and every terminal state \(x_f\), there exists an input \(u\) defined on a finite interval \([0,T]\) such that \(x(0) = x_0\) and \(x(T) = x_f\).
The above definition of controllability means that the state can be controlled to an arbitrary point no matter where the initial state is. This requirement is too strong for a real system because of diverse nonlinear effects. However, in practice, linear analysis may still be meaningful even when the objective is to understand complex nonlinear systems such as brain networks.
In the 2011 paper published in the Nature, controllability was deeply examined for many complex networks in the real world. That paper suggests that full control of about 80% of nodes in a gene regulatory network is required for controllability. Müller and Schuppert pointed out that this contradicts empirical results in the cellular reprogramming field. This gap was due to the conceptual difference of controllability in the two fields. In the context of biology, the target regions to be reached by control are only the local attractors that the system possess. Therefore, controlling of the state to an arbitrary point is not needed in biology.
Recently, Mochizuki's group developed a remarkable framework, called linkage logic, for controlling nonlinear dynamics in a biologically plausible sense. It is surprising that the reported framework is purely mathematical.
Suppose that the states \(x = (x_1,\ldots,x_n)\) of the considered system are governed by nonlinear ODEs
\[ \begin{align} \dot{x}_i(t) = f_i(x_1(t),\ldots,x_n(t)), \quad i \in \{1,\ldots,n\}. \label{eq:ODE} \end{align} \]
The structure of the interactions between states can be represented by a directed graph \(\mathcal{G} = (\mathcal{V},\mathcal{E})\). Each node corresponds to one of the states, and an edge between two nodes exists if the corresponding states interact. A key concept is a set of determining nodes, which is defined as follows.
Definition: A set of determining nodes is a subset \(I \subset \mathcal{V}\) such that if for every two solutions \(x,x^\prime\) to the nonlinear ODE \(\eqref{eq:ODE}\),
\[ \lim_{t \to \infty} \|x_i(t) - x_i^\prime(t)\| = 0, \quad i \in I, \]
implies
\[ \lim_{t \to \infty} \|x_i(t) - x_i^\prime(t)\| = 0, \quad i \in \mathcal{V}. \]
Mochizuki's group discovered that a set of determining nodes can be identified as a feedback vertex set, which is a concept from graph theory.
Another important object is a global attractor, which is the largest compact invariant set that attracts all trajectories. It is well known that if a dynamical system is dissipative in the sense of Levinson, then it has a unique global attractor. A mathematical explanation of Mochizuki's framework is as follows.
Theorem: If a subset \(I \subset \mathcal{V}\) is a feedback vertex set of \(\mathcal{G}\), then there exists an injection on the global attractor into the negative trajectories associated with \(I\).